# American Institute of Mathematical Sciences

November  2015, 35(11): 5185-5202. doi: 10.3934/dcds.2015.35.5185

## The obstacle problem for quasilinear stochastic PDEs with non-homogeneous operator

 1 Institut du Risque et de l'Assurance, Université du Maine, Avenue Olivier Messiaen, 72085 Le Mans Cedex 9, France, France 2 School of Mathematical Sciences, University of Fudan, Handan Road 220, 200433, Shanghai, China

Received  November 2012 Revised  May 2014 Published  May 2015

We prove the existence and uniqueness of solution of the obstacle problem for quasilinear Stochastic PDEs with non-homogeneous second order operator. Our method is based on analytical technics coming from the parabolic potential theory. The solution is expressed as a pair $(u,\nu)$ where $u$ is a predictable continuous process which takes values in a proper Sobolev space and $\nu$ is a random regular measure satisfying minimal Skohorod condition. Moreover, we establish a maximum principle for local solutions of such class of stochastic PDEs. The proofs are based on a version of Itô's formula and estimates for the positive part of a local solution which is non-positive on the lateral boundary.
Citation: Laurent Denis, Anis Matoussi, Jing Zhang. The obstacle problem for quasilinear stochastic PDEs with non-homogeneous operator. Discrete & Continuous Dynamical Systems - A, 2015, 35 (11) : 5185-5202. doi: 10.3934/dcds.2015.35.5185
##### References:
 [1] D. G. Aronson, On the Green's function for second order parabolic differential equations with discontinuous coefficients,, Bulletin of the American Mathematical Society, 69 (1963), 841.  doi: 10.1090/S0002-9904-1963-11059-9.  Google Scholar [2] D. G. Aronson, Non-negative solutions of linear parabolic equations,, Annali della Scuola Normale Superiore di Pisa, 22 (1968), 607.   Google Scholar [3] V. Bally and A. Matoussi, Weak solutions for SPDE's and Backward Doubly SDE's,, Journal of Theoret. Probab., 14 (2001), 125.  doi: 10.1023/A:1007825232513.  Google Scholar [4] P. Charrier and G. M. Troianiello, Un résultat d'existence et de régularité pour les solutions fortes d'un problème unilatéral d'évolution avec obstacle dépendant du temps,, C. R. Acad. Sci. Paris Sér. A-B, 281 (1975).   Google Scholar [5] L. Denis and L. Stoïca, A general analytical result for non-linear SPDE's and applications,, Electronic Journal of Probability, 9 (2004), 674.  doi: 10.1214/EJP.v9-223.  Google Scholar [6] L. Denis, A. Matoussi and L. Stoïca, $L^p$ estimates for the uniform norm of solutions of quasilinear SPDE's,, Probability Theory Related Fields, 133 (2005), 437.  doi: 10.1007/s00440-005-0436-5.  Google Scholar [7] L. Denis, A. Matoussi and L. Stoïca, Maximum Principle for Parabolic SPDE's: First Approach,, Stochastic Partial Differential Equations and Applications VIII in the series Quaderni di Matematica del Dipartimento di Matematica della Seconda Università di Napoli, (2011).   Google Scholar [8] L. Denis, A. Matoussi and L. Stoïca, Maximum principle and comparison theorem for quasi-linear stochastic PDE's,, Electronic Journal of Probability, 14 (2009), 500.  doi: 10.1214/EJP.v14-629.  Google Scholar [9] L. Denis and A. Matoussi, Maximum Principle for quasilinear SPDE's on a bounded domain without regularity assumptions,, Stochastic Processes and Their Applications, 123 (2013), 1104.  doi: 10.1016/j.spa.2012.10.005.  Google Scholar [10] L. Denis, A. Matoussi and J. Zhang, The obstacle problem for quasilinear stochastic PDEs: Analytical approach,, The Annals of Probability, 42 (2014), 865.  doi: 10.1214/12-AOP805.  Google Scholar [11] L. Denis, A. Matoussi and J. Zhang, Maximum principle for quasilinear SPDEs with obstacle,, Electronic Journal of Probability, 19 (2014), 1.  doi: 10.1214/EJP.v19-2716.  Google Scholar [12] C. Donati-Martin and E. Pardoux, White noise driven SPDEs with reflection,, Probability Theory Related Fields, 95 (1993), 1.  doi: 10.1007/BF01197335.  Google Scholar [13] N. El Karoui, C. Kapoudjian, E. Pardoux, S. Peng and M. C. Quenez, Reflected solutions of backward SDE and related obstacle problems for PDEs,, The Annals of Probability, 25 (1997), 702.  doi: 10.1214/aop/1024404416.  Google Scholar [14] N. El Karoui, S. Peng and M. C. Quenez, Backward stochastic differential equations in finance,, Mathematical Finance, 7 (1997), 1.  doi: 10.1111/1467-9965.00022.  Google Scholar [15] K. H. Kim, An Lp-theory of SPDEs of divergence form on Lipschitz domains,, Journal of Theoretical Probability, 22 (2009), 220.  doi: 10.1007/s10959-008-0170-x.  Google Scholar [16] T. Klimsiak, Reflected BSDEs and obstacle problem for semilinear PDEs in divergence form,, Stochastic Processes and their Applications, 122 (2012), 134.  doi: 10.1016/j.spa.2011.10.001.  Google Scholar [17] N. V. Krylov, An analytic approach to SPDEs,, in Stochastic Partial Differential Equations: Six Perspectives, (1999), 185.  doi: 10.1090/surv/064/05.  Google Scholar [18] N. V. Krylov, Maximum principle of SPDEs and its applications,, in Stochastic Differential Equations: Theory and Applications (eds. P. Baxendale and S. Lototsky), (2007), 311.  doi: 10.1142/9789812770639_0012.  Google Scholar [19] J. L. Lions and E. Magenes, Problèmes Aux Limites Non Homogènes et Applications,, Dunod, (1968).   Google Scholar [20] A. Matoussi and M. Xu, Sobolev solution for semilinear PDE with obstacle under monotonicity condition,, Electronic Journal of Probability, 13 (2008), 1035.  doi: 10.1214/EJP.v13-522.  Google Scholar [21] A. Matoussi and L. Stoïca, The obstacle problem for quasilinear stochastic PDE's,, The Annals of Probability, 38 (2010), 1143.  doi: 10.1214/09-AOP507.  Google Scholar [22] F. Mignot and J. P. Puel, Inéquations d'évolution paraboliques avec convexes dépendant du temps. Applications aux inéquations quasi-variationnelles d'évolution,, Arch. for Rat. Mech. and Ana., 64 (1977), 59.  doi: 10.1007/BF00280179.  Google Scholar [23] D. Nualart and E. Pardoux, White noise driven quasilinear SPDEs with reflection,, Probability Theory and Related Fields, 93 (1992), 77.  doi: 10.1007/BF01195389.  Google Scholar [24] M. Pierre, Problèmes d'Evolution avec Contraintes Unilaterales et Potentiels Parabolique,, Comm. in Partial Differential Equations, 4 (1979), 1149.  doi: 10.1080/03605307908820124.  Google Scholar [25] M. Pierre, Représentant précis d'un potentiel parabolique,, in Séminaire de Théorie du Potentiel, (1980), 186.   Google Scholar [26] F. Riesz and B. Nagy, Functional Analysis,, Dover, (1990).   Google Scholar [27] M. Sanz and P. Vuillermot, Equivalence and Hölder Sobolev regularity of solutions for a class of non-autonomous stochastic partial differential equations,, Ann. I. H. Poincaré, 39 (2003), 703.  doi: 10.1016/S0246-0203(03)00015-3.  Google Scholar [28] T. G. Xu and T. S. Zhang, White noise driven SPDEs with reflection: Existence, uniqueness and large deviation principles,, Stochatic Processes and Their Applications, 119 (2009), 3453.  doi: 10.1016/j.spa.2009.06.005.  Google Scholar

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##### References:
 [1] D. G. Aronson, On the Green's function for second order parabolic differential equations with discontinuous coefficients,, Bulletin of the American Mathematical Society, 69 (1963), 841.  doi: 10.1090/S0002-9904-1963-11059-9.  Google Scholar [2] D. G. Aronson, Non-negative solutions of linear parabolic equations,, Annali della Scuola Normale Superiore di Pisa, 22 (1968), 607.   Google Scholar [3] V. Bally and A. Matoussi, Weak solutions for SPDE's and Backward Doubly SDE's,, Journal of Theoret. Probab., 14 (2001), 125.  doi: 10.1023/A:1007825232513.  Google Scholar [4] P. Charrier and G. M. Troianiello, Un résultat d'existence et de régularité pour les solutions fortes d'un problème unilatéral d'évolution avec obstacle dépendant du temps,, C. R. Acad. Sci. Paris Sér. A-B, 281 (1975).   Google Scholar [5] L. Denis and L. Stoïca, A general analytical result for non-linear SPDE's and applications,, Electronic Journal of Probability, 9 (2004), 674.  doi: 10.1214/EJP.v9-223.  Google Scholar [6] L. Denis, A. Matoussi and L. Stoïca, $L^p$ estimates for the uniform norm of solutions of quasilinear SPDE's,, Probability Theory Related Fields, 133 (2005), 437.  doi: 10.1007/s00440-005-0436-5.  Google Scholar [7] L. Denis, A. Matoussi and L. Stoïca, Maximum Principle for Parabolic SPDE's: First Approach,, Stochastic Partial Differential Equations and Applications VIII in the series Quaderni di Matematica del Dipartimento di Matematica della Seconda Università di Napoli, (2011).   Google Scholar [8] L. Denis, A. Matoussi and L. Stoïca, Maximum principle and comparison theorem for quasi-linear stochastic PDE's,, Electronic Journal of Probability, 14 (2009), 500.  doi: 10.1214/EJP.v14-629.  Google Scholar [9] L. Denis and A. Matoussi, Maximum Principle for quasilinear SPDE's on a bounded domain without regularity assumptions,, Stochastic Processes and Their Applications, 123 (2013), 1104.  doi: 10.1016/j.spa.2012.10.005.  Google Scholar [10] L. Denis, A. Matoussi and J. Zhang, The obstacle problem for quasilinear stochastic PDEs: Analytical approach,, The Annals of Probability, 42 (2014), 865.  doi: 10.1214/12-AOP805.  Google Scholar [11] L. Denis, A. Matoussi and J. Zhang, Maximum principle for quasilinear SPDEs with obstacle,, Electronic Journal of Probability, 19 (2014), 1.  doi: 10.1214/EJP.v19-2716.  Google Scholar [12] C. Donati-Martin and E. Pardoux, White noise driven SPDEs with reflection,, Probability Theory Related Fields, 95 (1993), 1.  doi: 10.1007/BF01197335.  Google Scholar [13] N. El Karoui, C. Kapoudjian, E. Pardoux, S. Peng and M. C. Quenez, Reflected solutions of backward SDE and related obstacle problems for PDEs,, The Annals of Probability, 25 (1997), 702.  doi: 10.1214/aop/1024404416.  Google Scholar [14] N. El Karoui, S. Peng and M. C. Quenez, Backward stochastic differential equations in finance,, Mathematical Finance, 7 (1997), 1.  doi: 10.1111/1467-9965.00022.  Google Scholar [15] K. H. Kim, An Lp-theory of SPDEs of divergence form on Lipschitz domains,, Journal of Theoretical Probability, 22 (2009), 220.  doi: 10.1007/s10959-008-0170-x.  Google Scholar [16] T. Klimsiak, Reflected BSDEs and obstacle problem for semilinear PDEs in divergence form,, Stochastic Processes and their Applications, 122 (2012), 134.  doi: 10.1016/j.spa.2011.10.001.  Google Scholar [17] N. V. Krylov, An analytic approach to SPDEs,, in Stochastic Partial Differential Equations: Six Perspectives, (1999), 185.  doi: 10.1090/surv/064/05.  Google Scholar [18] N. V. Krylov, Maximum principle of SPDEs and its applications,, in Stochastic Differential Equations: Theory and Applications (eds. P. Baxendale and S. Lototsky), (2007), 311.  doi: 10.1142/9789812770639_0012.  Google Scholar [19] J. L. Lions and E. Magenes, Problèmes Aux Limites Non Homogènes et Applications,, Dunod, (1968).   Google Scholar [20] A. Matoussi and M. Xu, Sobolev solution for semilinear PDE with obstacle under monotonicity condition,, Electronic Journal of Probability, 13 (2008), 1035.  doi: 10.1214/EJP.v13-522.  Google Scholar [21] A. Matoussi and L. Stoïca, The obstacle problem for quasilinear stochastic PDE's,, The Annals of Probability, 38 (2010), 1143.  doi: 10.1214/09-AOP507.  Google Scholar [22] F. Mignot and J. P. Puel, Inéquations d'évolution paraboliques avec convexes dépendant du temps. Applications aux inéquations quasi-variationnelles d'évolution,, Arch. for Rat. Mech. and Ana., 64 (1977), 59.  doi: 10.1007/BF00280179.  Google Scholar [23] D. Nualart and E. Pardoux, White noise driven quasilinear SPDEs with reflection,, Probability Theory and Related Fields, 93 (1992), 77.  doi: 10.1007/BF01195389.  Google Scholar [24] M. Pierre, Problèmes d'Evolution avec Contraintes Unilaterales et Potentiels Parabolique,, Comm. in Partial Differential Equations, 4 (1979), 1149.  doi: 10.1080/03605307908820124.  Google Scholar [25] M. Pierre, Représentant précis d'un potentiel parabolique,, in Séminaire de Théorie du Potentiel, (1980), 186.   Google Scholar [26] F. Riesz and B. Nagy, Functional Analysis,, Dover, (1990).   Google Scholar [27] M. Sanz and P. Vuillermot, Equivalence and Hölder Sobolev regularity of solutions for a class of non-autonomous stochastic partial differential equations,, Ann. I. H. Poincaré, 39 (2003), 703.  doi: 10.1016/S0246-0203(03)00015-3.  Google Scholar [28] T. G. Xu and T. S. Zhang, White noise driven SPDEs with reflection: Existence, uniqueness and large deviation principles,, Stochatic Processes and Their Applications, 119 (2009), 3453.  doi: 10.1016/j.spa.2009.06.005.  Google Scholar
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